3D subsurface imaging

3D subsurface imaging

3D Imaging of Large Scale Buried Structure by 1D Inversion of Very Early Time Electromagnetic (VETEM) Data

Results of a simple and efficient method for 3D subsurface imaging of inhomogeneous background are presented on this site. These results include large-scale applications on data provided by the VETEM system of USGS. The method has a computational complexity that linearly increases with the size of the terrain to be studied.

The pressing reason for the use of a simple 1D scheme is its low computational cost, and its potential for on-site processing of real experimental data. Despite the simplicity of the scheme, we demonstrate that it will still return useful information on the buried objects. The simple 1D scheme also calls for a simple experimental setup consisting of only a transmitter and receiver pair operating in the time domain.

A 3D Verification/Demonstration

This is a 3D simulation of the VETEM measurement system traversing a square region of 7.0 m x 7.0 m. A 2D PEC object of size 2 m x 2 m is buried at a depth of 1.5 m, two opposite corners of which are located at (-1 m, -1 m) and (1 m, 1 m) in the plots. At each pixel, a 3D CGFFT solver solves the forward problem for a set of frequencies ranging from DC to 5 MHz. Thus, at each pixel, a time-domain waveform, i.e., the simulated VETEM measurement, is obtained by inverse Fourier transforming this frequency-domain solution. In the calibration and inversion processes, no frequency greater than 5 MHz is utilized to preserve the validity of the dipole model.

Simulated waveform

Because of the strong direct coupling in the opposite direction due to a 2 degrees receiver tilt, the scattering from the metal plate actually decreases the magnitude and slope of the field before 2.0 microseconds in the above figure.

At each point, the uniform 20 mS/m background is estimated by a method that relies on the fact that higher frequencies will not penetrate into the deeper regions and will thus contain negligible scattered field from buried objects. In order not to invalidate the dipole model, the estimation cannot be performed at very high frequencies. Still, the estimates closely approximate the true value almost everywhere.

Background conductivity estimates

The 1D solver performs a reconstruction at each pixel, taking into account the multiple scattering in the depth direction. In this case, we assumed 20 homogeneous layers of 25 cm each. Thus, the resulting reconstruction yields a linear conductivity reconstruction at each position as shown below. The metal object can be discerned around 1.2 m instead of the actual 1.5 m. The error is most probably due to the two neglected dimensions of multiple scattering, which result in stronger fields than the 1D solver anticipates.

1D reconstruction

Five planar cuts of the above data at 0.5, 1.0, 1.5, 2.0, and 2.5 m deep manifest that a high conductivity object is located around 1.0 m - 1.5 m.

Plane cuts

From this point on, the processing can be somewhat arbitrary. One approach is to report the depths where the ratio of the extremum of the linear conductivity profile to the background value is larger than a given value. Just employing the average as the threshold yields the following.

Planar cuts

Enhancements

Code has been written to visualize the results using the visualization toolkit (VTK). (In some of the earlier contours/plots, the regions that were not traversed are set to either 0 m or 4 m. Please ignore those regions.)
It is evident that the receiving antenna was not completely perpendicular. Especially, waveforms with "Lenz effect" tails at the tip of a negative ramp suggest that the receiver antenna is slightly tilted. The negative ramp is mostly due to the direct, vertical magnetic dipole (VMD), coupling. This has been taken into account in the processes of calibration and inversion.
The background is now computed more robustly by a method that relies on the fact that higher frequencies will not penetrate into deeper regions and will therefore contain a lot less scattered field.

Challenges/Problems

Background conductivity is not known. It used to be estimated by the early-time ramp slope and hence used to be an average, experimental value. That is, the slope may turn out to be large when a PEC object is nearby even though the background conductivity is not large. Recently, it is computed more robustly by a method that takes into account the fact that higher frequencies will not penetrate into the deeper regions and will therefore contain a lot less scattered field.
The tilt angle of the receiver is unknown. This is a major problem since it brings about a vertical magnetic dipole (VMD) primary field contribution as well as more negligible VMD reflected and scattered contributions. It is possible to measure the tilting by noting that all the magnetostatic, i.e., DC coupling is due to the primary field. This is not true when the scatterers are perfectly conducting, in which case they contribute to the DC component. The calibration code attempts to remove the VMD reflected and primary fields by estimating the tilt angle, usually in the range 1-4 degrees. Note that the raw waveforms would look more like the calibration waveform with a positive ramp if the receiver were tilted towards the transmitter, i.e., | \ instead of | /. On the other hand, the inverse code attempts to employ the weaker VMD scattered in its reconstruction; however, it is often so small that it is lost in numerical noise.
Calibration is not perfect. In the transfer function, there seems to be considerable capacitive effect in addition to the expected inductive behavior. Also, the system tends to kill the near-DC portion of the waveforms.
There is a phase delay between the simulated reflected and primary fields convolved with the actual ramp and the measured VETEM waveform after the system function deconvolution. The slightly varying phase needs to be experimentally determined and is location-dependent. This is a relatively negligible problem because the phase difference is often less than twenty nanoseconds.

Pit-9 Results

1. Traverses	Actual traverses over pit 9.
2. Contour	Contour plot of the whole region.
3. Contour	Zoomed contour of the southwest region.
4. Color Plot	Color plot of the southwest region.
5. Contour	Estimated background conductivities.
6. Waveform Processing	A set of raw VETEM waveforms and their processing.
7. Calibration	Calibration and resulting system functions.

Foundry Site Results

1. Greyscale Plots	Grey-scale plots that summarize the results.
2. Traverses	Actual traverses.
3. Contour	Contour plot of the whole region.
4. Contour	Estimated background conductivities.